We consider the fair division of indivisible items among $n$ agents with additive non-negative normalized valuations, with the goal of obtaining high value guarantees, that is, close to the proportional share for each agent. We prove that partitions where \emph{every} part yields high value for each agent are asymptotically limited by a discrepancy barrier of $Θ(\sqrt{n})$. Guided by this, our main objective is to overcome this barrier and achieve stronger individual guarantees for each agent in polynomial time. Towards this, we are able to exhibit an exponential improvement over the discrepancy barrier. In particular, we can create partitions on-the-go such that when agents arrive sequentially (representing a previously-agreed priority order) and pick a part autonomously and rationally (i.e., one of highest value), then each is guaranteed a part of value at least $\mathsf{PROP} - \mathcal{O}{(\log n)}$. Moreover, we show even better guarantees for three restricted valuation classes such as those defined by: a common ordering on items, a bound on the multiplicity of values, and a hypergraph with a bound on the \emph{influence} of any agent. Specifically, we study instances where: (1) the agents are ``close'' to unanimity in their relative valuation of the items -- a generalization of the ordered additive setting; (2) the valuation functions do not assign the same positive value to more than $t$ items; and (3) the valuation functions respect a hypergraph, a setting introduced by Christodoulou et al. [EC'23], where agents are vertices and items are hyperedges. While the sizes of the hyperedges and neighborhoods can be arbitrary, the influence of any agent $a$, defined as the number of its neighbors who value at least one item positively that $a$ also values positively, is bounded.

翻译：我们考虑在具有加性非负归一化估值函数的 $n$ 个智能体之间公平分配不可分割物品的问题，目标是获得高价值保证，即每个智能体获得的价值接近其比例份额。我们证明，当\emph{每个}部分都能为每个智能体带来高价值的分配方案，其渐近上界受限于 $\Theta(\sqrt{n})$ 的差异性障碍。基于此，我们的主要目标是在多项式时间内克服这一障碍，为每个智能体实现更强的个体保证。为此，我们能够展示相对于差异性障碍的指数级改进。具体而言，我们可以动态构建分配方案，使得当智能体按序到达（代表预先商定的优先级顺序）并自主且理性地（即选择价值最高的部分）选择一个部分时，每个智能体保证获得至少 $\mathsf{PROP} - \mathcal{O}{(\log n)}$ 的价值。此外，针对三类受限估值函数类别，我们展示了更优的保证，这些类别包括：物品上的共同序关系、价值重复次数的界限，以及具有任意智能体\emph{影响力}界限的超图。具体而言，我们研究以下情形：(1) 智能体对物品的相对估值“接近”一致——这是有序加性设定的一种推广；(2) 估值函数不会对超过 $t$ 个物品赋予相同的正值；(3) 估值函数遵循超图结构，该设定由 Christodoulou 等人 [EC'23] 引入，其中智能体为顶点，物品为超边。虽然超边和邻域的规模可以是任意的，但任意智能体 $a$ 的影响力（定义为与其共享至少一个共同正估值物品的邻居数量）是有界的。